A.2 速度と渦度の関係

渦度の回転をとると、連続の式を用いて次の関係が得られる。
$\displaystyle \nabla \times$$\displaystyle \mbox{\boldmath$\omega $}$$\displaystyle =\nabla \times (\nabla \times$$\displaystyle \mbox{\boldmath$v$}$$\displaystyle )=\nabla \,(\nabla \cdot$$\displaystyle \mbox{\boldmath$v$}$$\displaystyle )-\nabla ^2$$\displaystyle \mbox{\boldmath$v$}$$\displaystyle =-\nabla ^2$$\displaystyle \mbox{\boldmath$v$}$  (38)

$ x,y,z$成分はそれぞれ次のように書ける。
$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,u$ $\displaystyle =$ $\displaystyle \frac{\partial \omega _3}{\partial y}-\frac{\partial \omega _2}{\partial z}$ (39)
$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,v$ $\displaystyle =$ $\displaystyle \frac{\partial \omega _1}{\partial z}-\frac{\partial \omega _3}{\partial x}$ (40)
$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,w$ $\displaystyle =$ $\displaystyle \frac{\partial \omega _2}{\partial x}-\frac{\partial \omega _1}{\partial y}$ (41)

展開係数で考える。$ u$について、$ n\geq 1$のとき
$\displaystyle -\{(ir_xl)^2+(ir_ym)^2-(r_zn)^2\}\,\hat{u}$ $\displaystyle =$ $\displaystyle ir_ym\,\hat{\omega }_3-r_zn\,\hat{\omega }_2$  
  $\displaystyle =$ $\displaystyle ir_ym\cdot \frac{i\,(r_xl\,\hat{\omega }_1+r_ym\,\hat{\omega }_2)}{r_zn}-r_zn\,\hat{\omega }_2$  
  $\displaystyle =$ $\displaystyle -\,\frac{(r_xl)\,(r_ym)\,\hat{\omega }_1+\{(r_ym)^2+(r_zn)^2\}\,\hat{\omega }_2}{r_zn}$  

よって
$\displaystyle \hat{u}=-\,\frac{(r_xl)\,(r_ym)\,\hat{\omega }_1+\{(r_ym)^2+(r_zn)^2\}\,\hat{\omega }_2}{r_zn\,\{(r_xl)^2+(r_ym)^2+(r_zn)^2\}}$ (42)

$ n=0$のときは
$\displaystyle -\{(ir_xl)^2+(ir_ym)^2\}\,\hat{u}$ $\displaystyle =$ $\displaystyle ir_ym\,\hat{\omega }_3$  

より、次のようになる。
$\displaystyle \hat{u}=\frac{ir_ym\,\hat{\omega }_3}{(r_xl)^2+(r_ym)^2}$ (43)

$ \hat{v},\hat{w}$についても同様に考えると

$\displaystyle \hat{v}$ $\displaystyle =$ $\displaystyle \left\{\begin{array}{l}
\displaystyle \frac{\{(r_xl)^2+(r_zn)^2\}...
...2} \qquad \qquad \quad \qquad \qquad \quad \,\,\,\,(\,n=0\,)
\end{array}\right.$ (44)
$\displaystyle \quad$  
$\displaystyle \hat{w}$ $\displaystyle =$ $\displaystyle \left\{\begin{array}{l}
\displaystyle \frac{i\,(r_xl\,\hat{\omega...
...ad \qquad \qquad \qquad \qquad \qquad \qquad \quad (\,n=0\,)
\end{array}\right.$ (45)

SAITO Naoaki
2008-03-07